\textit{This section contains material covered by IA Groups.}

\subsection{Definitions}
A \textit{group} is a pair \( (G, \cdot) \) where \( G \) is a set and \( \cdot \colon G \times G \to G \) is a binary operation on \( G \), satisfying
\begin{itemize}
	\item \( a \cdot (b \cdot c) = (a \cdot b) \cdot c \);
	\item there exists \( e \in G \) such that for all \( g \in G \), we have \( g \cdot e = e \cdot g = g \); and
	\item for all \( g \in G \), there exists an inverse \( h \in G \) such that \( g \cdot h = h \cdot g = e \).
\end{itemize}
\begin{remark}
	\begin{enumerate}
		\item Sometimes, such as in IA Groups, a closure axiom is also specified.
		      However, this is implicit in the type definition of \( \cdot \).
		      In practice, this must normally be checked explicitly.
		\item Additive and multiplicative notation will be used interchangeably.
		      For additive notation, the inverse of \( g \) is denoted \( -g \), and for multiplicative notation, the inverse is instead denoted \( g^{-1} \).
		      The identity element is sometimes denoted \( 0 \) in additive notation and \( 1 \) in multiplicative notation.
	\end{enumerate}
\end{remark}
A subset \( H \subseteq G \) is a \textit{subgroup} of \( G \), written \( H \leq G \), if \( h \cdot h' \in H \) for all \( h, h' \in H \), and \( (H, \cdot) \) is a group.
The closure axiom must be checked, since we are restricting the definition of \( \cdot \) to a smaller set.
\begin{remark}
	A non-empty subset \( H \subseteq G \) is a subgroup of \( G \) if and only if
	\[
		a, b \in H \implies a \cdot b^{-1} \in H
	\]
\end{remark}
An \textit{abelian} group is a group such that \( a \cdot b = b \cdot a \) for all \( a, b \) in the group.
The \textit{direct product} of two groups \( G, H \), written \( G \times H \), is the group over the Cartesian product \( G \times H \) with operation \( \cdot \) defined such that \( (g_1, h_1) \cdot (g_2, h_2) = (g_1 \cdot_G g_2, h_1 \cdot_H h_2) \).

\subsection{Cosets}
Let \( H \leq G \).
Then, the \textit{left cosets} of \( H \) in \( G \) are the sets \( gH \) for all \( g \in G \).
The set of left cosets partitions \( G \).
Each coset has the same cardinality as \( H \).
Lagrange's theorem states that if \( G \) is a finite group and \( H \leq G \), we have \( \abs{G} = \abs{H} \cdot [G \colon H] \), where \( [G \colon H] \) is the number of left cosets of \( H \) in \( G \).
\( [G \colon H] \) is known as the \textit{index} of \( H \) in \( G \).
We can construct Lagrange's theorem analogously using right cosets.
Hence, the index of a subgroup is independent of the choice of whether to use left or right cosets; the number of left cosets is equal to the number of right cosets.

\subsection{Order}
Let \( g \in G \).
If there exists \( n \geq 1 \) such that \( g^n = 1 \), then the least such \( n \) is the \textit{order} of \( G \).
If no such \( n \) exists, we say that \( g \) has infinite order.
If \( g \) has order \( d \), then:
\begin{enumerate}
	\item \( g^n = 1 \implies d \mid n \);
	\item \( \genset{g} = \qty{1, g, \dots, g^{d-1}} \leq G \), and by Lagrange's theorem (if \( G \) is finite) \( d \mid \abs{G} \).
\end{enumerate}

\subsection{Normality and quotients}
A subgroup \( H \leq G \) is \textit{normal}, written \( H \trianglelefteq G \), if \( g^{-1} H g = H \) for all \( g \in G \).
In other words, \( H \) is preserved under conjugation over \( G \).
If \( H \trianglelefteq G \), then the set \( \faktor{G}{H} \) of left cosets of \( H \) in \( G \) forms the \textit{quotient group}.
The group action is defined by \( g_1 H \cdot g_2 H = (g_1 \cdot g_2) H \).
This can be shown to be well-defined.

\subsection{Homomorphisms}
Let \( G, H \) be groups.
A function \( \phi \colon G \to H \) is a \textit{group homomorphism} if \( \phi(g_1 \cdot_G g_2) = \phi(g_1) \cdot_H \phi(g_2) \) for all \( g_1, g_2 \in G \).
The \textit{kernel} of \( \phi \) is defined to be \( \ker \phi = \qty{g \in G \colon \phi(g) = 1} \), and the \textit{image} of \( \phi \) is \( \Im \phi = \qty{\phi(g) \colon g \in G} \).
The kernel is a normal subgroup of \( G \), and the image is a subgroup of \( H \).

\subsection{Isomorphisms}
An \textit{isomorphism} is a homomorphism that is bijective.
This yields an inverse function, which is of course also an isomorphism.
If \( \varphi \colon G \to H \) is an isomorphism, we say that \( G \) and \( H \) are isomorphic, written \( G \cong H \).
Isomorphism is an equivalence relation.
The isomorphism theorems are
\begin{enumerate}
	\item if \( \varphi \colon G \to H \), then \( \faktor{G}{\ker \varphi} \cong \Im \varphi \);
	\item if \( H \leq G \) and \( N \trianglelefteq G \), then \( H \cap N \trianglelefteq H \) and \( \faktor{H}{H \cap N} \cong \faktor{HN}{N} \);
	\item if \( N \leq M \leq G \) such that \( N \trianglelefteq G \) and \( M \trianglelefteq G \), then \( \faktor{M}{N} \trianglelefteq \faktor{G}{N} \), and \( \faktor{G/N}{M/N} = \faktor{G}{M} \).
\end{enumerate}
